1. Field of the Invention
The present invention relates to a differential amplifier circuit comprised of bipolar transistors and more particularly, to a differential amplifier circuit having an improved transconductance linearity within a wide input voltage range, which is formed on a bipolar semiconductor integrated circuit device and is operable at a low supply voltage.
2. Description of the Prior Art
A differential amplifier circuit having a superior transconductance linearity within a comparatively wide input voltage range has been known as an "Operational Transconductance Amplifier (OTA)".
An example of the conventional bipolar OTAs was disclosed by Schmook in the IEEE Journal of Solid-State Circuits, Vol. SC-10, No. 6, PP. 407-411, December 1975, in which two unbalanced differential pairs of bipolar transistors are employed. Two transistors of each pair has different emitter areas or sizes. Output ends or collectors of the two transistors of each pair are cross-coupled.
The transconductance linearization technique proposed by Schmook is known as the "Multi-tanh" technique, and has been practically used.
An application of this Multi-tanh technique was disclosed y Tanimoto et al., in IEEE Journal of Solid-State Circuits, Vol. 26, No. 7, PP. 937-945, July 1991.
FIG. 1 shows a conventional bipolar OTA of this sort, which contains first to (2N+1)-th differential pairs of npn bipolar transistors and first to (2N+1)-th constant current sinks.
As shown in FIG. 1, a first differential pair is made of npn bipolar transistors Q01 and Q02 whose emitter areas are equal to that of a unit bipolar transistor. Emitters of the transistors Q01 and Q02 are commonly connected to one end of a first constant current sink CS0 (current: I.sub.00). The other end of the current sink CS0 is grounded. Bases of the transistors Q01 and Q02 are connected to first and second input terminals T1 and T2, respectively.
A second differential pair is made of npn bipolar transistors Q11 and Q12 whose emitter areas are different. The emitter area of the transistors Q11 is equal to that of a unit bipolar transistor. The emitter area of the transistors Q12 is K.sub.1 times as much as that of a unit bipolar transistor, where K.sub.1 is greater than unity. Emitters of the transistors Q11 and Q12 are commonly connected to one end of a second constant current sink CS1 (current: I.sub.01) The other end of the current sink CS1 is grounded. Bases of the transistors Q11 and Q12 are connected to the first and second input terminals T1 and T2, respectively.
A third differential pair is made of npn bipolar transistors Q11' and Q12' whose emitter areas are different. The emitter area of the transistor Q11' is equal to that of a unit bipolar transistor. The emitter area of the transistors Q12' is K.sub.1 times as much as that of a unit bipolar transistor. Emitters of the transistors Q11' and Q12' are commonly connected to one end of a third constant current sink CS1' (current: I.sub.01) The other end of the current sink CS1' is grounded. Bases of the transistors Q11' and Q12' are connected to the first and second input terminals T2 and T1, respectively.
Collectors of the transistors Q11 and Q12' are coupled together. A collector of the transistor Q01 is connected to the coupled collectors of the transistors Q11 and Q12'. Collectors of the transistors Q11' and Q12 are coupled together. A collector of the transistor Q02 is connected to the coupled collectors of the transistors Q11' and Q12.
Similarly, a 2N-th differential pair is made of npn bipolar transistors QN1 and QN2 whose emitter areas are different. The emitter area of the transistor QN1 is equal to that of a unit bipolar transistor. The emitter area of the transistors QN2 is K.sub.N times as much as that of a unit bipolar transistor, where K.sub.N is greater than unity. Emitters of the transistors QN1 and QN2 are commonly connected to one end of an 2N-th constant current sink CSN (current: I.sub.ON). The other end of the current sink CSN is grounded. Bases of the transistors QN1 and QN2 are connected to the first and second input terminals T1 and T2, respectively.
A (2N+1)-th differential pair is made of npn bipolar transistors QN1' and QN2' whose emitter areas are different. The emitter area of the transistor QN1' is equal to that of a unit bipolar transistor. The emitter area of the transistors QN2' is K.sub.N times as much as that of a unit bipolar transistor. Emitters of the transistors QN1' and QN2' are commonly connected to one end of an (2N+1)-th constant current sink CSN' (current: I.sub.ON). The other end of the current sink CSN' is grounded. Bases of the transistors QN1' and QN2' are connected to the first and second input terminals T2 and T1, respectively.
Collectors of the transistors QN1 and QN2' are coupled together. Collectors of the transistors QN1' and QN2 are coupled together.
The collector of the transistor Q1 and the coupled collectors of the transistors Q11, Q12', . . . , QN1' and QN2' are further coupled together, thereby forming one of differential output ends of this conventional OTA. The collector of the transistor Q2 and the coupled collectors of the transistors Q12, Q11', . . . , QN2 and QN1' are coupled together, thereby forming the other of the differential output ends of this conventional OTA.
A differential input voltage V.sub.1 as an input signal to be amplified is applied across the first and second input terminals T1 and T2. A differential output current .DELTA.I as an amplified output signal is derived from the differential output ends.
Here, it is supposed that all the transistors Q01 through QN2' are matched in characteristic and that the base-width modulation (i.e., the Early voltage) is ignored.
For a j-th differential pair of two bipolar emitter-coupled transistors which are driven by a common constant current I.sub.oj and which have emitter areas in the ratio of K.sub.j : 1 (K.sub.j is equal to or greater than unity) with respect to a unit bipolar transistor, collector currents I.sub.C1j and I.sub.C2j of the two transistors are expressed as the following equations (1a) and (1b), respectively. ##EQU1##
In the equations (1a) and (1b), V.sub.BE1j and V.sub.BE2j are base-to-emitter voltages of the two transistors, respectively, and I.sub.s the saturation current thereof. V.sub.T is the thermal voltage defined as V.sub.T =kT/q, where k is the Boltzmann's constant, T is absolute temperature in degrees Kelvin, and q is the charge of an electron.
When the differential input voltage V.sub.1 is applied across the bases of the two transistors of the j-th differential pair, the following equation (1c) is established around the loop consisting of the input voltage and the two base-emitter junctions because of the Kirchhoff's voltage law. EQU V.sub.i -V.sub.BE1j +V.sub.BE2j =0 (1c)
The equation (1c) is rewritten by substituting the above equations (1a) and (1b) into the equation (1c) as the following equation (1d): ##EQU2##
On the other hand, the following relationship (1e) is satisfied among the collector currents I.sub.C1j and I.sub.C2j and the constant driving current I.sub.0j. EQU I.sub.C1j +I.sub.C2j =.alpha..sub.F I.sub.0j ( 1e)
where .alpha..sub.F is a dc common-base current gain factor.
Therefore, the collector currents I.sub.C1j and I.sub.C2j are expressed as the following equations (1f) and (1g), respectively: ##EQU3##
Accordingly, a differential output current .DELTA.I.sub.Cj of the j-th differential pair, which is defined as .DELTA.I.sub.Cj =I.sub.C1j -I.sub.C2j, is given by the following equation (2) or (3). ##EQU4##
In the equations (2) and (3), V.sub.Kj is a dc offset voltage between the two transistors of the j-th differential pair, which is defined as V.sub.Kj =V.sub.r .multidot.ln (K.sub.j).
It is seen from the equations (2) and (3) that the differential output current .DELTA.I.sub.Cj of the j-th differential pair is expressed by a hyperbolic tangent (tanh) function, in other words, by a fraction whose numerator is a hyperbolic sine (sinh) function and whose denominator is a hyperbolic cosine (cosh) function.
Accordingly, a differential output current .DELTA.I of the conventional OTA shown in FIG. 1 is expressed as the following equation (4a) or (4b): ##EQU5##
In the first differential pair of the transistors Q01 and Q02, the emitter areas of the transistors Q01 and Q02 are equal to that of a unit transistor and therefore, the offset voltage V.sub.K0 is zero, because V.sub.K0 =V.sub.T .multidot.ln(1)=0.
The name of the "multi-tanH" technique was termed after the form of the equation (4a).
Here, since the offset voltages of the (j-1)-th and j-th differential pairs, which are arranged symmetrically, have the same emitter area ratio, they have a relationship of V.sub.Kj =-V.sub.Kj-1. Therefore, the differential output currents .DELTA.I.sub.Cj and .DELTA.I.sub.Cj-1 of these two pairs are expressed as the following equations (5a) and (5b), respectively. ##EQU6##
The collectors of the transistors of the symmetrically-arranged (j-1)-th and j-th differential pairs are coupled together and accordingly, the following equation (6) is established as ##EQU7##
Then, the differential output current .DELTA.I of the conventional OTA of FIG. 1 is expressed as the following equation (7). ##EQU8## where .DELTA.I.sub.c0 is the differential output current of the first differential pair of the transistors Q01 and Q02.
As a result, the differential output current .DELTA.I of the conventional OTA of FIG. 1 is expressed as the following general equation (8). ##EQU9## where B.sub.j and C.sub.j are coefficients.
If the equation (8) is differentiated by the differential input voltage V.sub.i, the transconductance characteristic of the conventional OTA of FIG. 1 is given by the following equation (9). ##EQU10##
To cause the transconductance to be maximally flat, considering the symmetric arrangement of the differential transistor pairs, the odd-order differential coefficients of the differential output current .DELTA.I in the equation (8) needs to be zero at V.sub.i =0. Specifically, the following condition (10) needs to be satisfied. ##EQU11##
The maximum value of the transconductance shown in the equation (9) is obtained at V.sub.i =0. Therefore, the following equation (11) needs to be established. ##EQU12##
FIG. 2 shows another conventional bipolar OTA of this sort, which contains first to N-th differential pairs of bipolar transistors and first to N-th constant current sinks
As shown in FIG. 2, this OTA has the same configuration as that of the OTA of FIG. 1 except that the first differential pair of the transistors Q01 and Q02 and the corresponding first constant current sink CS0 are canceled.
Accordingly, a differential output current .DELTA.I of the conventional OTA shown in FIG. 2 is expressed as the following equation (7'), because .DELTA.I.sub.Co =0 in the equation (7): ##EQU13##
As a result, by substituting the above equation (6) into the equation (7'), the differential output current .DELTA.I of the conventional OTA of FIG. 2 is expressed as the following general equation (8'). ##EQU14##
If the equation (8') is differentiated by the differential input voltage V.sub.i, the transconductance characteristic of the conventional OTA of FIG. 2 is given by the following equation (9'). ##EQU15##
To cause the transconductance to be maximally flat, considering the symmetric arrangement of the differential transistor pairs, the odd-order differential coefficients of the differential output current .DELTA.I in the equation (8') needs to be zero at V.sub.i =0. The maximum value of the transconductance shown in the equation (9') is obtained at V.sub.i =0. Therefore, also in the conventional OTA of FIG. 2, the above condition (10) needs to be satisfied and the above equation (11) needs to be established.
When N=1 (i.e., 2N=2) in the conventional OTA of FIG. 2, this OTA has a circuit configuration shown in FIG. 3. This configuration is termed a "multi-tanh doublet".
The condition of the transconductance to be maximally flat is given by ##EQU16##
Therefore, the emitter area ratio K.sub.1 is given as ##EQU17## and the coefficients B.sub.0, B.sub.1 and C.sub.1 are given as B.sub.0 =cosh (ln K.sub.1)=2
B.sub.1 =1 and PA1 C.sub.1 =2. PA1 cosh(V.sub.K1 /V.sub.T)=1.5 (i.e., K.sub.1 =expcosh.sup.-1 V.sub.K1 !.apprxeq.2.6180) PA1 cosh(V.sub.K2 /V.sub.T)=4 (i.e., K.sub.2 =expcosh.sup.-1 V.sub.K2 !.apprxeq.7.873), PA1 B.sub.2 =1, B.sub.1 =16, B.sub.0 =18, C.sub.2 =4, and C.sub.1 =22.
A differential output current .DELTA.I of the conventional OTA of FIG. 3 is expressed as the following equation (12): ##EQU18##
When N=1 (i.e., 2N+1=3) in the conventional OTA of FIG. 1, this OTA has a circuit configuration shown in FIG. 5. This configuration is termed a "multi-tanh triplet".
A differential output current .DELTA.I of the conventional OTA shown in FIG. 5 is expressed as the following equation (13): ##EQU19##
The condition of the transconductance to be maximally flat is given by ##EQU20##
Therefore, the following equation (14) is established. ##EQU21##
Further, from the relationship of ##EQU22## the following equation (15) is obtained. ##EQU23##
Therefore, cosh(V.sub.K1 /V.sub.r)=2 (i.e., K.sub.1 .apprxeq.7.873) and I.sub.00 =(16/25) I.sub.01 are established. This means that B2=1, B1=2coshln (K.sub.i)+1!=9, B0=0, C2=2.64 and C1=6.48.
As a result, the differential output current .DELTA.I is given by the following equation (16), where I.sub.0 =I.sub.01. ##EQU24##
When N=2 (i.e., 2N=4) in the conventional OTA of FIG. 2, this configuration is termed a "multi-tanh quin".
A differential output current .DELTA.I of this conventional OTA is expressed as the following equation (17): ##EQU25##
Since the condition of the transconductance to be maximally flat is given by ##EQU26## the following equation (18) is obtained. ##EQU27##
From the condition that the fifth-order differential coefficient of the differential output current .DELTA.I is equal to zero at V.sub.i =0, i.e., ##EQU28## the following equation (19) is obtained. ##EQU29##
Therefore, ##EQU30## are established.
Thus, the constant current I.sub.01 is given as ##EQU31##
Accordingly, B.sub.2 =1, B.sub.1 =16, B.sub.0 =18, C.sub.2 .apprxeq.3.0957 I.sub.01, and C.sub.1 .apprxeq.19.81252 I.sub.01 are obtained.
The differential output current .DELTA.I is given by the following equation (20), where I.sub.0 =I.sub.01. ##EQU32##
For the above multi-tanh quintuplet, various transfer functions can be obtained from the condition of ##EQU33##
For example, when I.sub.01 =I.sub.02 =I.sub.0,
the coefficients are given as
The differential output current .DELTA.I in this case is expressed by the following equation (21). ##EQU34##
The input voltage range of the multi-tanh quintuplet providing a linear transconductance characteristic, which is given by the equation (21), is approximately as narrow as that of the multi-tanh triplet given by the above equation (16).
Generally, to cause the input voltage range to be as wide as possible in the multi-tanh cells having maximally flat characteristics, high, odd-order differential coefficients (a fifth-order differential coefficient or higher) needs to be zero at V.sub.i =0.
It should be noted that the denominator of each differential output current .DELTA.I expressed by the above equation (12), (16), (17) or (20) is expressed by a sum of hyperbolic cosine functions and that each of the coefficients of the hyperbolic cosine functions are expressed by a natural number.
With the above described conventional OTAs employing the multi-tanh technique, an obtainable input voltage range providing the linear transconductance characteristic is at most 200 mV.sub.p-r.
FIG. 4 represents the transconductance characteristics of the multi-tanh doublet of FIG. 3 (curve c), the multi-tanh triplet of FIG. 5 (curve b) and the multi-tanh quad (curve a), in which V.sub.I is the thermal voltage whose value is approximately 25 to 26 mV at room temperature).
As described above, with the conventional OTAs using the multi-tanh technique, the denominator of the differential output current .DELTA.I is expressed by a cosh function or functions and the numerator thereof by a sinh function or functions.
When the numerator of the differential output current .DELTA.I includes a sinh function or functions only, the transconductance linearity is dependent upon the functional form of the denominator thereof.
An OTA is an essential, basic function block in analog signal applications. Recently, fabrication processes for large-scale integrated circuit devices (LSIs) have been becoming finer and finer and as a result, the supply voltage for the LSIs has been decreasing from 5 V to 3 V, 2V, or 1V. This tendency has been increasing the necessity for the low-voltage circuit technique more and more.
Also, although the above-described conventional OTAs using the multi-tanh technique are capable of low-voltage operation, they have a problem that the input voltage range providing the linear transconductance characteristic is very narrow. To cause the input voltage range to be wider, another problem that the circuit scale and current consumption increase will occur.
Further, with the above-described conventional OTAs using the multi-tanh technique, the dc offset voltage for each differential transistor pair is generated by the difference of the emitter areas or sizes of the corresponding two bipolar transistors, a realizable emitter-area ratio of the transistors is at most several tens. As a result, a problem that an obtainable input voltage range is at most approximately 100 mV.sub.P-P occurs.